Clothoids

A clothoid is a curve $(x(s),y(x))$ is described by the parametric equations:

$$ x(s)=\int_0^s \cos\left(\frac{1}{2}\kappa'\tau^2+\kappa_0\tau+\vartheta_0\right),\mathrm{d}\tau $$

$$ y(s)=\int_0^s \sin\left(\frac{1}{2}\kappa'\tau^2+\kappa_0\tau+\vartheta_0\right),\mathrm{d}\tau $$

when $\kappa'=0$ the clothoids reduce to a circle arc and when $\kappa'=\kappa_0=0$ the clothoids reduce to a straight segment.

This library implements algorithms for $G^1$ and $G^2$ fitting with clothoids, spline of clothoids, circle arc and biarc.

The implementation of the algorithms on clothoids, splines of clothoids, arc, bi-arc splines of biarcs are described in the works:

• E.Bertolazzi, M.Frego, $G^1$ fitting with clothoids, Mathematical Methods in the Applied Sciences, 2015, https://doi.org/10.1002/mma.3114

• E.Bertolazzi, M.Frego, Interpolating clothoid splines with curvature continuity, Mathematical Methods in the Applied Sciences, 2018, https://doi.org/10.1002/mma.4700

• E.Bertolazzi, M.Frego, On the G2 Hermite interpolation problem with clothoids, Journal of Computational and Applied Mathematics, 2018, https://doi.org/10.1016/j.cam.2018.03.029

• E.Bertolazzi, M.Frego, A Note on Robust Biarc Computation, Computer-Aided Design And Applications, 2019, https://doi.org/10.14733/cadaps.2019.822-835

• E.Bertolazzi, M.Frego, Point-Clothoid Distance and Projection Computation, SIAM Journal on Scientific Computing, 2019, https://doi.org/10.1137/18M1200439

• E.Bertolazzi, M.Frego, Francesco Biral, Interpolating splines of biarcs from a sequence of planar points, Computer-Aided Design And Applications, 2020, https://doi.org/10.14733/cadaps.2021.66-85

• E.Bertolazzi, P.Bevilacqua, M.Frego, Efficient intersection between splines of clothoids, Mathematics and Computers in Simulation, 2020, https://doi.org/10.1016/j.matcom.2019.10.001

• E.Bertolazzi, M.Frego, Francesco Biral, Interpolating splines of biarcs from a sequence of planar points, Computer-Aided Design And Applications, 2020, https://doi.org/10.14733/cadaps.2021.66-85

• M.Frego, Closed form parametrisation of 3D clothoids by arclength with both linear varying curvature and torsion, Applied Mathematics and Computation, 2022, https://doi.org/10.1016/j.amc.2021.126907

The library contains the following objects:

• Segment
• CircleArc
• Clothoids
• BiArc
• spline of
  • Segment
  • CircleArc
  • Clothoids (with $G^1$ and $G^2$ continuity)
  • BiArc
• Triangles
• BBox (bounding box)

and fast algorithms involving the objects, in particular:

• evaluation
• intersection (between objects)
• point-object distance

Library is written in C++11 with a MATLAB mex interface. Thus can be used in fast compiled application or in MATLAB scripts.

Compilation

To compile the C++11 library the easy way require cmake and rake

ruby setup.rb

then

rake

to build the MATLAB toolbox

cd toolbox
ruby populate_toolbox.rb
ruby build.rb

for more details see: online documentation at http://ebertolazzi.github.io/Clothoids/